Pinchoff by surface diffusion

Abstract

We construct surface diffusion flows $f:{\mathbb{T}}^2\times [0,T)\to {\mathbb{R}}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in [0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $\mu >0$ and smooth function $U:\mathbb{R}\to \mathbb{R}$ such that \[ A(t)=\{4μ(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)ζ,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(ζ). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.

48 pages, 1 figure